It is commonplace to consider applications of mathematics to other fields, especially the exact sciences. But what I would like to know about is the converse topic, namely:
What are some applications of other
fields to mathematics?
Obviously the applications of physics to mathematics are ubiquitous (gauge theory is just one significant modern example, and quantum algorithms and mirror symmetry are others…the list from physics goes on). For the purposes of this question (at least) theoretical computer science is just a branch of mathematics.
So answers involving fields other than physics are of particular interest to me (and answers involving theoretical computer science are of little to no interest to me), as are answers where the application isn't bidirectional (for example, one could say that game theory is an application of mathematics to economics as much if not more than an application of economics to mathematics).
Finally (at least for the purposes of this question), anything of the form "phenomenon Y was experimentally observed and it turned out that there was a rich but hitherto unknown mathematical theory Z explaining Y" is not that interesting as an application to mathematics unless the discovery of Z has some truly special status. Something like (e.g.) symplectic geometry might fall under this (leaving aside the "experimental" bit), but is not of particular interest for reasons above.
Best Answer
According to Gordan, the Hilbert Basis Theorem was an application of theology.